On an eigenvector-dependent nonlinear eigenvalue problem from the perspective of relative perturbation theory
Let $H(V)$ be an $n\times n$ Hermitian matrix-valued function of $V$, an orthogonal $n\times k$ complex matrix, for integers $n$, $k$ and $k\leq n$. The eigenvector-dependent nonlinear eigenvalue problem (NEPv) $H(V)V=V\Lambda,$ where $\Lambda$ is a $k\times k$ complex matrix, was studied by several authors, see, e.g., [\textit{Y. Cai} et al., SIAM J. Matrix Anal. Appl. 39, No. 3, 1360--1382 (2018; Zbl 1401.65036)]. The authors of this paper use perturbation theory (see [\textit{R. Bhatia}, Matrix analysis. New York, NY: Springer (1996; Zbl 0863.15001); \textit{G. W. Stewart} and \textit{J.-g. Sun}, Matrix perturbation theory. Boston etc.: Academic Press, Inc. (1990; Zbl 0706.65013)]) and results from [\textit{R. S. Ismagilov}, Dokl. Akad. Nauk SSSR 149, 769--772 (1963; Zbl 0204.15903); \textit{C. Davis} and \textit{W. M. Kahan}, SIAM J. Numer. Anal. 7, 1--46 (1970; Zbl 0198.47201)] to obtain sufficient conditions on the solvability and uniqueness of the solution of the above NEPv.
eigenvector-dependent nonlinear eigenvalue problem
scientific article; zbMATH DE number 7355002
scientific article